Catenary Sag

Galileo thought a hanging chain was a parabola, and said so in the Two New Sciences in 1638. He was wrong, and the correction took half a century: Jungius doubted it, and in 1691, in response to a challenge posed by Jakob Bernoulli, three independent solutions arrived from Huygens, Leibniz and Johann Bernoulli. The curve is the hyperbolic cosine, and Huygens gave it the name catenaria, from the Latin for chain.

But Galileo was not simply confused, and this is the part most textbooks fumble. The parabola is the exact shape for a load spread evenly along the horizontal, and the catenary is the exact shape for a load spread evenly along the arc. Neither is an approximation to the other. A suspension bridge's main cable, holding up a level deck through closely spaced hangers, really is close to a parabola, because the deck is what weighs anything and the deck is laid out along the ground. A bare chain is a catenary, because the chain's mass follows the chain. Galileo was describing the wrong load case, not making an arithmetic error, and a suspension bridge proves he was describing a real one.

The two nevertheless agree remarkably well when the cable is shallow, and the reason is visible in the algebra. Expand $a(\cosh(L/2a) - 1)$ as a series and the leading term is $L^{2}/(8a)$ — which, since $a = H/w$, is exactly $wL^{2}/(8H)$, the parabolic sag. The next correction goes as $L^{4}/a^{3}$. When $L$ is small against $a$ that correction is negligible and the two curves are indistinguishable; when it is not, it grows as the fourth power and the gap opens fast. Below a sag ratio of about 1 in 10 the difference is under 1%, which is why the parabola is the working tool of practice and the catenary is reserved for deep sags, long spans and the cases where the last fraction of a percent is money.

The catenary's price is that it does not invert. Given $a$ and $L$ the sag comes out in one line. Going the other way, from a measured sag and span to the parameter, is impossible in closed form: $a$ sits inside the hyperbolic cosine and also multiplies the whole expression, and no rearrangement separates those two appearances. Every sag-tension program on earth solves it by Newton iteration, converging in three or four steps from a parabolic first guess. This site does not iterate behind your back and hand you the result labelled as a formula, so that direction has no page. The span direction, by contrast, does invert exactly — $L$ appears only inside the cosh, and the inverse hyperbolic cosine undoes it — so that one is here.

Two limitations to keep in front of you. This assumes both supports at the same level, so the low point sits at midspan. On an inclined span the low point slides toward the lower support, may fall outside the span altogether on a steep one, and the two support tensions differ — the uphill dead-end always governs, because it is the point highest above the directrix. And this is still-air geometry at a single temperature. A conductor sags further when it is hot, further again under ice, and swings under wind, so a real sag-tension study runs this same shape at every design condition and designs for the worst of them.